A Liouville-type result for some non-cooperative Fisher--KPP systems and nonlocal equations on cylinders

Léo Girardin (1) , Quentin Griette (2)

1 UP11 UFR Sciences - Université Paris-Sud - Paris 11 - Faculté des Sciences
2 IMB - Institut de Mathématiques de Bordeaux

We address the uniqueness of the nonzero stationary state for a reaction-diffusion system of Fisher-KPP type that does not satisfy the comparison principle. Although the uniqueness is false in general, it turns out to be true under biologically natural assumptions on the parameters. This Liouville-type result is then used to characterize the long-time behavior of traveling waves. All results are extended to an analogous nonlocal reaction-diffusion equation that contains as a particular case the cane toads equation with bounded traits.

Mots clés: Liouville-type result; traveling wave; KPP nonlinearity; reaction--diffusion system; cane toads equation

Domaines: Equations aux dérivées partielles [math.AP]

Léo Girardin, Quentin Griette. A Liouville-type result for some non-cooperative Fisher--KPP systems and nonlocal equations on cylinders. 2020. ⟨hal-02406343v2⟩